Timothy Dimond

A Review of Tilting Pad Bearing Theory

A theoretical basis for static and dynamic operation of tilting pad journal bearings (TPJBs) has evolved over the last 50 years. Originally demonstrated by Lund using the pad assembly method and a classic Reynolds equation solution, the current state of the art includes full thermoelastohydrodynamic solutions of the generalized Reynolds equation that include fluid convective inertia effects, pad motions; and thermal and mechanical deformations of the pads and shaft. The development of TPJB theory is reviewed, emphasizing dynamic modeling. The paper begins with the early analyses of fixed geometry bearings and continues to modern analyses that include pad motion and stiffness and damping effects. The development of thermohydrodynamic, thermoelastohydrodynamic, and bulk-flow analyses is reviewed. The theories of TPJB dynamics, including synchronous and nonsynchronous models, are reviewed. A discussion of temporal inertia effects in tilting pad bearing is considered. Future trends are discussed, and a path for experimental verification is proposed.

Modal Tilt/Translate Control and Stability of a Rigid Rotor with Gyroscopics on Active Magnetic Bearings

Most industrial rotors supported in active magnetic bearings (AMBs) are operated well below the first bending critical speed. Also, they are usually controlled using proportional, integral and derivative controllers, which are set up as modally uncoupled parallel and tilt rotor axes. Gyroscopic effects create mode splitting and a speed-dependent plant. Two AMBs with four axes of control must simultaneously control and stabilize the rotor/AMB system. Various analyses have been published considering this problem for different rotor/AMB configurations. There has not been a fully dimensionless analysis of these rigid rotor AMB systems. This paper will perform this analysis with a modal PD controller in terms of translation mode and tilt mode dimensionless eigenvalues and eigenvectors. The number of independent system parameters is significantly reduced. Dimensionless PD controller gains, the ratio of rotor polar to transverse moments of inertia and a dimensionless speed ratio are used to evaluate a fully general system stability rigid rotor analysis. An objective of this work is to quantify the effects of gyroscopics on rigid rotor AMB systems. These gyroscopic forces reduce the system stability margin. The paper is also intended to help provide a common framework for communication between rotating machinery designers and controls engineers

Nonlinear Analysis of Squeeze Film Damper with Entrained Air in Rotordynamic Systems

Squeeze film dampers are widely used in aircraft engines, land-based gas turbines, and other rotating machines to improve system damping. They often have entrained air within the oil film, which is usually not taken into account in rotordynamics analysis due to lack of good formulations of the effects involved. The effects on the damping force calculation can be significant. This work presents a new formulation of the nonlinear Reynolds equation pressure evaluation within the squeeze film damper, including the effects of entrained air, with resulting changes in effective lubricant density and viscosity. Viscosity and density expressions are developed as a function of the air/oil volume fraction. The density of the bubbly oil is a function of the air bubble diameter, which changes due to surface tension effects during lubricant motion in the bubbly oil film. The lubricant viscosity decreases due to the entrained volume of air but increases due to the surface tension effects taken from experimental tests. Pressure supply and bubbly oil film cavitation effects are included in the analysis and end seal effects are evaluated. The nonlinear time-transient forces in the squeeze film damper are evaluated as functions of (1) lubricant and air properties; (2) damper geometry including diameter, length, clearance; (3) end seal properties; and (4) shaft position and velocity. Example cases of pressure calculations and radial and tangential forces are shown. Example nonlinear transient motions are presented for a rigid, symmetrical rotor and for a nonsymmetrical rotor representing a gas turbine–type fan rotor.

Comparison of Tilting-Pad Journal Bearing Dynamic Full Coefficient and Reduced Order Models Using Modal Analysis (GT2009-60269)

There is a significant disagreement in the literature concerning the proper evaluation of the experimental identification and frequency response of tilting-pad journal bearings (TPJBs) due to shaft excitations. Two linear models for the frequency dependence of TPJBs have been proposed. The first model, the full coefficient or stiffness-damping (KC) model, considers N p tilting pads and two rotor radial motions for N p +2 degrees of freedom. The dynamic reduction of the KC model results in eight frequency-dependent stiffness and damping coefficients. The second model, based on bearing system identification experimental results, employs 12 frequency-independent stiffness, damping, and mass (KCM) coefficients; pad degrees of freedom are not considered explicitly. Experimental data have been presented to support both models. There are major differences in the two approaches. The present analysis takes a new approach of considering pad dynamics explicitly in a state-space modal analysis. TPJB shaft and bearing pad stiffness and damping coefficients are calculated using a well known laminar, isothermal analysis and a pad assembly method. The TPJB rotor and pad KC model eigenvalues and eigen-vectors are then evaluated using state-space methods, with rotor and bearing pad inertias included explicitly in the model. The KC model results are also nonsynchronously reduced to the eight stiffness and damping coefficients and are expressed as shaft complex impedances. The system identification method is then applied to these complex impedances, and the state-space modal analysis is applied to the resulting KCM model. The damping ratios, natural frequencies, and mode shapes from the two bearing representations are compared. Two sample TPJB cases are examined in detail. The analysis indicated that four underdamped modes, two forward and two backward, dominate the rotor response over excitation frequencies from 0 to approximately running speed. The KC model predicts additional nearly critically damped modes primarily involving pad degrees of freedom, which do not exist in the identified KCM model. The KCM model results in natural frequencies that are 63-65% higher than the KC model. The difference in modal damping ratio estimates depends on the TPJB considered; the KCM estimate was 7-17% higher than the KC model. The results indicate that the KCM system identification method results in a reduced order model of TPBJ dynamic behavior, which may not capture physically justifiable results. Additionally, the differences in the calculated system natural frequency and modal damping have potential implications for rotordynamic analyses of flexible rotors.

Properties and Performance of Gas-Expanded Lubricants in Tilting Pad Journal Bearings

Lubricants enable proper function and reduce friction in rotating machinery, but they can also contribute to power loss and heat buildup. Gas-expanded lubricants (GELs) have been proposed as tunable mixtures of lubricant and CO2 under pressure with properties such as viscosity that can be controlled directly in response to changing environmental or rotordynamic conditions. In this work, experimental results of GEL viscosity, gas diffusivity, and thermal conductivity were combined with high-pressure phase equilibrium data to understand how these mixtures will behave in tilting pad journal bearings under a range of industry-relevant high-speed conditions. Simulations were carried out using the experimental data as inputs to a thermoelastohydrodynamic model of tilting pad journal bearing performance. Viscosity could be easily tuned by controlling the composition of the GEL and the effect on bearing efficiency was appreciable, with 14–46% improvements in power loss. This trend held for a range of lubricant chemistries with polyalkylene glycols, polyalpha olefins, and a polyol ester tested in this work. Diffusivity, which drives how readily CO2 and lubricants form homogenous mixtures, was found to be a function of the viscosity of the synthetic lubricant, with more viscous lubricants having a lower diffusivity than less viscous formulations. Model results for a bearing in a pressurized housing suggested that cavitation would be minimal for a range of speed conditions. Other bearing parameters, such as eccentricity, temperature, and minimum film thickness were relatively unchanged between conventionally lubricated and GEL-lubricated bearings, suggesting that the efficiency improvements could be achieved with few performance tradeoffs.

Time Transient Analysis of Horizontal Rigid Rotor Supported With O-Ring Sealed Squeeze Film Damper

This study addresses the nonlinear dynamic behavior of O-ring seals as the retaining spring in squeeze film dampers (SFDs). An analytical model is developed to predict the restoring and hysteresis forces of elastomer O-rings based on experimental and numerical data. This model takes into account the temperature softening and excitation frequency hardening effects in O-rings as well as the installation conditions in the form of radial and vertical preloads, σ and γ, respectively. Long bearing assumption is adopted for the solution of Reynolds equation. The equations of motion of horizontal unbalanced rigid rotor are derived, and a dimensional analysis is conducted on them. The numerical results substantiates the synchronizing effects of bearing parameter, B and vertical preload, γ, and the asynchronizing effects of O-ring parameter, O and radial preload, σ. It is shown that the variation of temperature and rotational speed as operating conditions influence the rotor response significantly.Copyright

Control of Active Magnetic Bearing systems on non-static foundations

A case study of an electric motor with Active Magnetic Bearings (AMBs) that is subjected to the waving of the supporting foundation is presented. The motor is intended for sea applications where the ship is disturbed by the oscillation of the waves. Initially, a tilt-and-translate PID controller is designed to stabilize the rotor in the magnetic bearing system and satisfy the Zone A specifications defined by the International Organization for Standardization (ISO 14839) for new machines with magnetic bearings. The performance of the tilt-and-translate controller is then tested when the motor is subjected to the waving of the foundation. Simulation shows that the rotor vibration level easily exceeds the permitted level by the ISO specifications due to a combination of the waving disturbance from the foundation and the gyroscopic forces. Based on the results with the tilt-and-translate controller, a robust μ controller is designed to reduce the vibration of the rotor. The performance of the controller is tested when subjected to the waving of the foundation, and simulation shows that the vibration level is reduced to the level permitted by the ISO specifications.

Comparison of Tilting-Pad Journal Bearing Dynamic Full Coefficient and Reduced Order Models Using Modal Analysis

There is significant disagreement concerning the frequency response of tilting pad journal bearings (TPJBs) due to non-synchronous excitations. Two linear models for the frequency dependence of TPJBs have been proposed. The first model, the full-coefficient or KC model, considers Np tilting pads and rotor motions for Np + 2 degrees of freedom. Dynamic reduction of the KC model results in eight frequency-dependent stiffness and damping coefficients. The second model, based on results from bearing system identification experiments, yields twelve frequency-independent stiffness, damping, and mass (KCM) coefficients. Experimental data has been presented to support both models. There are major differences in the two approaches. The analysis in this paper takes a new approach of considering the pad dynamics explicitly in a state-space modal analysis. TPJB shaft and bearing pad stiffness and damping coefficients are calculated using a well known laminar, isothermal analysis and a pad assembly method. The TPJB rotor and pad full system eigenvalues and eigenvectors are then evaluated using state-space methods, with rotor and bearing pad inertias included explicitly in the model. The full bearing coefficient results are also non-synchronously reduced to the 8 stiffness and damping coefficients are and expressed as shaft complex impedances. The system identification method is then applied to these complex impedances, and the state space modal analysis is applied to the resulting KCM model. The damping ratios, natural frequencies, and mode shapes from the two bearing representations are compared. Two example TPJBs are examined in detail. The analysis indicated that four underdamped modes, two forward and two backward, dominate the rotor response over excitation frequencies from 0 to running speed. The full coefficient, non-synchronously reduced model predicts additional critically damped or overdamped modes due to the additional degrees of freedom as compared to the identified KCM model. The KCM model results in natural frequencies that are 63–65 percent higher than the full coefficient model. The difference in modal damping ratio estimates depend on the TPJB considered, with KCM being 7–17 percent higher than the full coefficient model. The full coefficient model also indicates that the bearing pads contribute significantly to the underdamped modes. The results indicate that the system identification method results in a reduced order model of TPBJ dynamic behavior. Additionally, the differences in the modal calculated system natural frequency and modal damping have potential implications for rotordynamic analyses of flexible rotors, such as critical speed and stability analyses.Copyright

Calculation of Dynamic Coefficients for a Magnetically Levitated Artificial Heart Pump Using a CFD Approach

The artificial heart community acknowledges the 3rd generation Ventricular Assist Devices (VADs) as the leading technology in mechanical blood pump development. This category consists of rotary pumps with no mechanical or fluid bearings in contact with the fluid medium, usually magnetic or noncontacting hydrodynamic bearings. A magnetic suspension prevents the rotating impeller from contacting the pump’s internal surfaces and reduces regions of stagnant and high shear flow that normally surround a fluid or mechanical bearing. Magnetic bearings have no moving parts in contact and thus do not wear over time; this generally lengthens the operational life of the pumps as compared to those supported by conventional bearings. Employing this 3rd generation technology, the University of Virginia has been developing a ventricular assist device (LifeFlow) with a rotor that is suspended entirely by magnetic bearings. In order to perform the stability analysis, the hydrodynamic effects of the rotating impeller should be included in the calculation. This study describes the method to calculate the stiffness, damping, and mass coefficients, based on the CFD prediction of radial fluid forces exerted on the impeller due to its eccentric position inside the pump housing over a range of operating conditions. In consideration of the suspension design, the fluid forces exerted on the levitated axial impeller were estimated using CFD such that any fluid perturbations would be accounted for and counterbalanced during the suspension and motor design phase.Copyright

Coupled Lateral and Torsional Nonlinear Transient Rotor–Bearing System Analysis With Applications

This paper provides a time transient method for solving coupled lateral and torsional analysis of a flexible rotor–bearing system including gyroscopic effects, nonlinear short journal bearings, nonlinear short squeeze film dampers (SFDs), and external nonlinear forces/torques. The rotor is modeled as linear, and the supporting components, including bearings and dampers, are modeled as nonlinear. An implicit Runge–Kutta method is developed to solve the nonlinear equations of motion with nonconstant operating speed since the unbalance force and the gyroscopic effect are related to both the rotational speed and the acceleration. The developed method is compared with a previous torsional analysis first to verify the nonlinear transient solver. Then the coupled lateral and torsional analysis of an example flexible three-disk rotor, perhaps representing a compressor, with nonlinear bearings and nonlinear dampers driven by a synchronous motor is approached. The acceleration effects on lateral and torsional amplitudes of vibration are presented in the analysis. The developed method can be used to study the rotor motion with nonconstant rotational speed such as during startup, shutdown, going through critical speeds, blade loss force, or other sudden loading.